NOT
THE KNITTING YOU KNOW

Daina Taimina

Because hyperbolic space has
infinitely many symmetries, you can sculpt out of them whatever you want.
The series in this show demonstrates how different the same shape can
look.

Artist's statement

I was born in Riga, Latvia.
All my formal education is in mathematics and theoretical computer science.
I got my PhD degree for thesis "Behavior of Different Types of Automata
and Turing Machines on Infinite Words" in 1990. My thesis advisor was
Professor Rusins Freivalds, now well known name in quantum computing.
For 20 years I was teaching in University of Latvia, where most of my
students were prospective mathematics teachers. I was also leading numerous
workshops for teachers and editing math textbooks in Latvia. But one of
my hobbies always was knitting that I learned in middle school. I learned
how to crochet on my own because I liked to use crochet to finish my knitting
projects.

I was seeing patterns and algorithms
in knitting and crochet but I was not connecting it with my professional
work in mathematics until 1997 when I became a Visiting Associate Professor
in Department of Mathematics, Cornell University. Once I was participating
in geometry workshop led by Professor David Henderson. He was showing
a paper model of hyperbolic plane that was made using William Thurston's
idea of annuli. And then it came in my mind - if one can make it out of
a paper, then I should be able to crochet it and to get a more durable
model to use in my geometry class. In that fall I was assigned to teach
Math 451, using Henderson's book "Experiencing Geometry". During summer
1997 I made my first classroom set of hyperbolic planes and used it in
my geometry class. It was amazing to see how much it helped my students
to understand the nature of hyperbolic plane.

Since then I have made numerous
models of hyperbolic plane, including crocheting figures for 2nd and 3rd
edition of "Experiencing Geometry". For the last one I am now co-author
with my husband David Henderson, who always has some idea what I could
crochet next to use in his or my classes.

In 2003 I started to exchange
e-mails with Margaret Wertheim about how to crochet these models (she
learned about them from New Scientist), and in May 2004 The Institute
for Figuring invited us to give a public lecture "Crocheting the Hyperbolic
Plane" (see www.theiff.org for more
information). For classroom models I use acrylic yarn because it is more
durable and easy to clean. For this exhibit I did some of these shapes
in wool to stress that we can find hyperbolic shapes in nature, we are
just not used to notice them as easy as we do with flat or spherical shapes
that are other common surfaces with constant curvature.

We all know positive numbers,
negative numbers and zero. For surfaces we can use a notion of curvature
- zero curvature is flat surface, sphere has constant positive curvature.
It is logical to ask - but what surface has a negative constant curvature
and how does it look? The answer is hyperbolic plane and looking at the
models you can see how differently it can appear and how amazingly it
can be sculpted starting from the same basic shape.